# Ngô Quốc Anh

## December 31, 2011

### A Hardy-Moser-Trudinger inequality: A conjecture by Wang and Ye

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 21:48

Let $B$ denote the standard unit disk in $\mathbb R^2$. The famous Moser–Trudinger inequality says that

$\displaystyle\int_B {\exp \left( {\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}}} \right)dx} \leqslant C < \infty ,\quad\forall u \in H_0^1(B)\backslash \{ 0\}$

holds. There is another important inequality in analysis, the Hardy inequality which claims that

$\displaystyle H(u) = \int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} \geqslant 0,\quad\forall u \in H_0^1(B)$

holds. The one $H$ is usuall called the Hardy functional. One can immediately see that

$\displaystyle\frac{{4\pi {u^2}}}{{\left\| {\nabla u} \right\|_2^2}} \leqslant \dfrac{{4\pi {u^2}}}{{\displaystyle\int_B {|\nabla u{|^2}dx} - \int_B {\frac{{{u^2}}}{{{{(1 - |x{|^2})}^2}}}dx} }}$

for any $u \in H_0^1(B)\backslash \{ 0\}$. Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant $C_0 >0$ such that the following

$\displaystyle\int_B {\frac{{4\pi {u^2}}}{{H(u)}}dx} \leqslant C_0 < \infty ,\quad\forall u \in \mathcal H(B^n)\backslash \{ 0\}$

where $B^n$ is the unit ball in $\mathbb R^n$, $n \geqslant 2$ and $\mathcal H=\mathcal H(B^n)$ is the complement of $C_0^\infty(B^n)$ with respect to the following norm $\|u\|_{\mathcal H}=\sqrt{H(u)}$.

Let us go back to the case $n=2$. They then defined

$\displaystyle {H_d}(u) = \int_\Omega {|\nabla u{|^2}dx} - \frac{1}{4}\int_\Omega {\frac{{{u^2}}}{{d{{(x,\partial \Omega )}^2}}}dx} > 0,\quad \forall u \in H_0^1(\Omega )\backslash \{ 0\}$

where $\Omega$ is a regular, bounded and convex domain sitting in $\mathbb R^2$. They then conjectured that the following

$\displaystyle\int_\Omega {\frac{{4\pi {u^2}}}{{{H_d}(u)}}dx} \leqslant C(\Omega ) < \infty ,\quad\forall u \in {\mathcal H_d}(\Omega )\backslash \{ 0\}$

still holds for some constant $C(\Omega)>0$ where ${\mathcal H_d}(\Omega )$ denotes the completion of $C_0^\infty (\Omega)$ with the corresponding norm associated with $H_d$. Apparently, the conjecture holds true for $\Omega = B$.

## August 16, 2010

### The Moser-Trudinger inequality for domains with holes

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:42

In this entry, we are interested in the following result

Theorem (Moser-Trudinger’s inequality for domains with holes). Let $\Omega$ be a bounded smooth domain in $\mathbb R^2$. Let $S_1$ and $S_2$ be two subsets of $\overline \Omega$ satisfying

${\rm dist}(S_1,S_2) \geqslant \delta_0>0$

and let $\gamma_0$ be a number satisfying $\gamma_0 \in \left(0,\frac{1}{2}\right)$. Then for any $\varepsilon>0$, there exists a constant $c=c(\varepsilon, \delta_0, \gamma_0)>0$ such that

$\displaystyle\int_\Omega {{e^u}} \leqslant C\exp \left[ {\frac{1}{{32\pi - \varepsilon }}\int_\Omega {{{\left| {\nabla u} \right|}^2}} + C} \right]$

holds for all $u \in H_0^1(\Omega)$ satisfying

$\displaystyle\frac{{\int_{{S_1}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}, \quad \frac{{\int_{{S_2}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}$.

## August 13, 2010

### An application of the (Moser-)Trudinger inequality to the mean field equations

Let $(M,g)$ be a compact Riemannian surface with the volume $|M|$. The simplest form of the mean field equation studied in the contexts of the prescribing Gaussian curvature, statistical mechanics of many vortex points in the perfect fluid and self-dual gauss theories is given by

$\displaystyle - {\Delta _g}u = \lambda \left( {\frac{{{e^u}}}{{\int_M {{e^u}d{v_g}} }} - \frac{1}{{|M|}}} \right), \quad \text{ on } M$

with

$\displaystyle\int_M {ud{v_g}} = 0$

where $\lambda$ is a real number.

The mean field equation has a variational structure, and $u$ is a solution if and only if it is a critical point of

$\displaystyle {J_\lambda }(v) = \frac{1}{2}\int_M {{{\left| {\nabla v} \right|}^2}d{v_g}} - \lambda \log \int_M {{e^v}d{v_g}}$

defined for $v \in H^1(M)$ with

$\displaystyle\int_M {vd{v_g}} = 0$.

It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for $\mathbb S^2$

Theorem (Moser-Trudinger’s inequality for $\mathbb S^2$). There are constants $\eta>0$ and $c=c(g)>0$ such that for each $p \geqslant 2$

$\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}$

for all $u \in W^{1,2}(\mathbb S^2)$.

## July 8, 2010

### The Moser-Trudinger inequality

Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.

Let us remind the Trudinger inequality

Theorem (Trudinger). Let $\Omega \subset \mathbb R^n$ be a bounded domain and $u \in W_0^{1, n}(\Omega)$ with

$\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1$.

Then there exist universal constants $\beta>0$, $C_1>0$ such that

$\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega |$.

The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant $\beta_0$. In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows

Theorem (Moser). Let $\Omega \subset \mathbb R^n$ be a bounded domain and $u \in W_0^{1, n}(\Omega)$ with

$\displaystyle\int_\Omega {{{\left| {\nabla u} \right|}^n}dx} \leqslant 1$.

Then there exist sharp constants $\beta_0=\beta(n)>0$, $C_1=C_1(n)>0$ given by

$\displaystyle \beta_0=n\omega_{n-1}^{\frac{1}{n}-1}$

such that

$\displaystyle\int_\Omega {\exp (\beta {|u|^\frac{n}{n-1}})dx} \leqslant {C_1}|\Omega|, \quad \forall \beta \leqslant \beta_0$.

The constant $\beta_0$ is sharp in the sense that for all $\beta>\beta_0$ there is a sequence of functions $u_k \in W_0^{1,n}(\Omega)$ satisfying

$\displaystyle\int_\Omega {{{\left| {\nabla u_k} \right|}^n}dx} \leqslant 1$

but the integral

$\displaystyle\int_\Omega {\exp (\beta {|u_k|^\frac{n}{n-1}})dx}$

grow without bound.

For general compact closed manifold $(M,g)$ the constant on the right hand side of the Moser-Trudinger inequality depends on the metric $g$. Working on a sphere $(\mathbb S^2,g_c)$ with a canonical metric allows us to control the constants.

Theorem (Moser). There is a universal constant $C_1>0$ such that for all $u \in W^{1,2}(\mathbb S^2)$ with

$\displaystyle\int_{\mathbb S^2} {{{\left| {\nabla u} \right|}^n}dv_{g_c}} \leqslant 1$

and

$\displaystyle\int_{\mathbb S^2} u dv_{g_c}=0$

we have

$\displaystyle\int_{\mathbb S^2} \exp(4\pi u^2) \leqslant {C_1}$.

Observe that

$\displaystyle 4\pi = \int_{{\mathbb{S}^2}} {d{v_{{g_c}}}} <\int_{{\mathbb{S}^2}} {{e^{4\pi {u^2}}}d{v_{{g_c}}}} \leqslant {C_1}$.

In the same way as we introduce in the entry concerning the Trudinger inequality one can show

Corollary. For

$\displaystyle C_2 := \log C_1 +\log\frac{1}{4\pi}$

one has

$\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}$

for all $u \in W^{1,2}(\mathbb S^2)$.

Obviously, $C_2 >0$ since $C_1 >4\pi$. It turns out to determine the best constant $C_2$. This had been done by Onofri known as the Onofri inequality [here].

Theorem (Onofri).Let $u \in W^{1,2}(\mathbb S^2)$ then we have

$\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u$

with the equality iff

$\Delta u +e^{2u}=1$.

The proof of the Onofri inequality relies on a result due to Aubin

Theorem (Aubin). For all $\varepsilon>0$ there exists a constant $C_\varepsilon$ such that

$\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\left( {\frac{1}{2} + \varepsilon } \right)\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_\varepsilon }$

for any $u$ belonging to the following class

$\displaystyle S = \left\{ {u \in {W^{1,2}}({\mathbb{S}^2}):\int_{{\mathbb{S}^2}} {{e^{2u}}{x_j}d{v_{{g_c}}}} = 0,j = \overline {1,3} } \right\}$.

Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.

## July 1, 2010

### The Trudinger inequality

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 10:53

In 1967, Neil S. Trudinger announced a result in J. Math. Mech. (now known as Indiana Univ. Math. J.) which can be seen as a limiting case of the Sobolev inequality [here] or [here].

It is well-known from the Sobolev embedding theorem that

$W_0^{\alpha, q}(\Omega) \hookrightarrow L^p(\Omega)$

for

$\displaystyle \frac{1}{p}=\frac{1}{q}-\frac{\alpha}{n}, \quad q\alpha.

The case $q\alpha=n$ is commonly referred to the limitting case. If $\alpha=1$, $n=2$ and $q<2$ we obtain

$W_0^{1, q}(\Omega) \hookrightarrow L^p(\Omega)$.

In general one cannot take the limits $q \to 2$ and $p \to \infty$, i.e.

$W_0^{1, 2}(\Omega) \not\hookrightarrow L^\infty(\Omega)$.

A counter-example is given by

$\displaystyle \log\left(1+\log\frac{1}{|x|}\right)$